Asthma, allergic rhinitis, and eczema are all statistically more prevalent in children of households with animals present than without. This was a hospital-based study that looked at Qatari children, so the mix of animals (cats, goats, birds) weren't exactly the same mix as you'd find in, say, the USA, but the statistical significance was very strong (p=0.008 to p=0.0001)

This study of Dutch children was a little more interesting, as it looked at both current and past pet ownership (dogs, cats, rodents, or birds). People who have always had pets (and still do) were actually the healthiest, followed by people who had never had pets. So if you stopped reading the paper there, you might think "oh, maybe having pets is actually good for your respiratory health!" But the researchers also asked about respiratory symptoms (asthma, coughing, wheezing, etc.) in people who used to have pets but no longer have pets...and often, those households no longer have pets because the children had developed respiratory issues. And some of the people who never had pets made that choice precisely because they had pre-existing respiratory issues that they did not want to aggravate. So it's not enough to look just at who currently has animals, because people change their behavior over time, and sometimes in response to the very variables you're trying to study.

In short, yes, it appears that living with animals (especially cats) increases the risk of developing respiratory symptoms like asthma, allergies, wheezing, etc...and that those symptoms persist at a higher rate than experienced by non-animal-owners even if the household goes animal-free.